Question

Solve the equation 3x+x²=-7. Does the number of solutions match the information provided by the discriminant? Explain

Answer 1

`-(3)/(2)+(√(19)i)/(2)` and
`-(3)/(2)-(√(19)i)/(2)`

Explanation:

`3x+x^2=-7\\\Rightarrow x^2+3x+7=0`

Discriminant

`D=b^2-4ac\\\Rightarrow D=3^2-4* 7* 1\\\Rightarrow D=-19`

If the discriminant is a negative number then the roots of the equation are imaginary

`x=(-b\pm \sqrtD)/(2a)\\\Rightarrow x=(-3\pm √(-19))/(2* 1)\\\Rightarrow x=-(3)/(2)\pm (√(19)i)/(2)\\\Rightarrow x=-(3)/(2)+(√(19)i)/(2), x=-(3)/(2)-(√(19)i)/(2)`

So, the two solutions are
`-(3)/(2)+(√(19)i)/(2)` and
`-(3)/(2)-(√(19)i)/(2)`